What positive integer n has the property that n² and n³ together contain every digit from 0 to 9 exactly once?

(In reply to Analytic attempt...? by tomarken)

Because there is a set pattern to the digital roots of n^2, n^3, and (n^2+n^3), n must be equal to 3+9k, 6+9k, 8+9k, or 9+9k (such that k is a positive integer). The digital root of a number composed of one each of the ten decimal digits is 9.
With this information, of the 12 numbers remaining from the sieve,
four more may be eliminated {47, 77, 59, 83} in the analytic attempt. The potential candidates now whittled down to 8:
53, 54, 62, 63, 69, 72, 84, 87

Posted by Dej Mar on 2021-04-07 16:02:23